Question

Marty sketches triangle DEF on the coordinate plane. He then sketches triangle D'E'F' on the same place. What is the scale factor going from DEF to D'E'F'?

D (0,0)
E (6,-3)
F (9,6)

D' (0,0)
E' (4,-2)
F' (6,4)


a. 3/2
b. 2/3
c. 3
d. 2

Answer 1

To find the scale factor going from triangle DEF to triangle D'E'F', we can calculate the lengths of corresponding sides in both triangles and then determine the ratio.

Let's first calculate the length of side DE in triangle DEF:

The coordinates of D and E are:
- D (0, 0)
- E (6, -3)

Using the distance formula:
\[
DE = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(6 - 0)^2 + (-3 - 0)^2} = \sqrt{6^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
\]

Next, let's calculate the length of side D'E' in triangle D'E'F':
- D' (0, 0)
- E' (4, -2)

Using the distance formula:
\[
D'E' = \sqrt{(4 - 0)^2 + (-2 - 0)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]

Now we find the scale factor by dividing the lengths of the corresponding sides:
\[
\text{Scale factor} = \frac{D'E'}{DE} = \frac{2\sqrt{5}}{3\sqrt{5}} = \frac{2}{3}
\]

Thus, the scale factor going from triangle DEF to triangle D'E'F' is \(\frac{2}{3}\).

The answer is:
**b. 2/3**